Multiple-input multiple-output (MIMO) is a family of techniques that utilize multiple antennas at the transmitter or at the receiver, or at both the transmitter and the receiver, to exploit the spatial dimension in order to improve data throughput and transmission reliability. The data throughput can be increased by either spatial multiplexing or beamforming.
Spatial multiplexing allows multiple data streams to be transmitted simultaneously to the same user through parallel channels in the MIMO setting, especially for diversity antennas where spatial correlation between antennas (both at the transmitter and the receiver) is low. Beamforming helps to enhance the signal-to-interference-plus-noise ratio (SINR) of the channel, thereby improving the channel rate. Such SINR improvement is achieved by proper weighting over multiple transmit antennas. The weight calculation can be based on either long-term measurement (e.g., open-loop) or via feedback (e.g., closed-loop). Closed-loop transmit weighting is often called precoding in the context of MIMO study.
FIG. 1 illustrates a precoded MIMO for a single user (SU) where M data streams, u1, . . . , uM, are spatially multiplexed by exploiting the M by N spatial channel matrix H. Since the number of transmitter antennas N is greater than the number of receiver antennas M, precoding is applied which is denoted as the matrix F.
Precoded MIMO can also be operated in multi-user MIMO (MU-MIMO) mode to further improve the sum rate across multiple users sharing the same time and frequency resource. FIG. 2 illustrates a two-user MU-MIMO where beamforming (e.g., precoding) is used to spatially separate the two users (and improving SINR), while for each user the two data streams (light shaded and dark shaded) are spatially multiplexed.
MU-MIMO, especially the downlink MU-MIMO, is a hot topic in Third Generation Partnership Project (3GPP) Long Term Evolution-Advanced (LTE-Advanced) study as described in 3GPP TR 36.814, v1.1.1, “Further Advancements for E-UTRA, Physical Layer Aspects,” June 2009. MU-MIMO may further enhance the data throughput of LTE systems. The Work Item of DL MU-MIMO was created in 3GPP Physical Layer Working Group (RAN1).
A key specification-impacting aspect of precoded MIMO is the spatial CSI feedback required for closed-loop precoding. The spatial channel matrix H as seen in FIG. 1 contains the complete spatial CSI. Alternatively, an N-by-N covariance matrix R, represented asR=HHH  (1)can provide sufficient spatial information for transmitter precoding where the superscript “H” denotes the complex conjugate. In general, it is too costly to feed back the floating-point version of H or R, which usually contains quite a number of complex coefficients in each frequency band. Quantization is therefore needed to make the feedback more efficient.
A codebook, known to both the receiver and the transmitter, is often used for CSI quantization so that only a codeword index is fed back. The codeword can be selected to either maximize the channel capacity or minimize the distance between the floating-point CSI and the quantized CSI.
Codebook design itself is a research-rich topic since a good codebook has to efficiently span the entire relevant spatial space. In that sense, generic codebooks are seldom efficient and, practically, codebooks are tailored to fit different antenna configurations and deployment scenarios. Generally speaking, the more complex the antenna configuration is, the more difficult the codebook design.
Table 1 is an excerpt from 3GPP RAN1 LTE standard specification described in 3GPP TS 36.211, “Evolved Universal Terrestrial Radio Access (E-UTRA); Physical Channels and Modulation”. The codebook is used for a very simple MIMO configuration with two transmit and two receiver antennas, M=2 and N=2, as in FIG. 1. As such, the maximum number of multiplexed streams (also called the layers) is 2.
TABLE 1A codebook in LTE specification for 2 × 2 MIMO.CodebookNumber of layers υindex120      1          2        ⁡      [                            1                                      1                      ]        1          2        ⁡      [                            1                          0                                      0                          1                      ]   1      1          2        ⁡      [                            1                                                  -            1                                ]        1    2    ⁡      [                            1                          1                                      1                                      -            1                                ]   2      1          2        ⁡      [                            1                                      j                      ]        1    2    ⁡      [                            1                          1                                      j                                      -            j                                ]   3      1          2        ⁡      [                            1                                                  -            j                                ]  —
Compared to single-user MIMO (SU-MIMO), multi-user MIMO (MU-MIMO) requires more accurate spatial CSI feedback in order to perform effective spatial separation and multiplexing operations. As a result, the CSI feedback and the codebook design in MU-MIMO are more challenging.
In mathematics, a Kronecker product, denoted by , is an operation on two matrices of arbitrary size resulting in a block matrix. For example,
                              A          ⊗          B                =                              [                                                                                                      a                      11                                        ⁢                    B                                                                    …                                                                                            a                                              1                        ⁢                        n                                                              ⁢                    B                                                                                                ⋮                                                  ⋱                                                  ⋮                                                                                                                        a                                              m                        ⁢                                                                                                  ⁢                        1                                                              ⁢                    B                                                                    …                                                                                            a                      mn                                        ⁢                    B                                                                        ]                    .                                    (        2        )            
The Kronecker product has been used in codebook design, for example for cross-polarization antennas described in 3GPP, R1-094752, “DL codebook design for 8Tx MIMO in LTE-A,” ZTE, RAN#59, Jeju, South Korea, November 2009 More specifically, the codebook is constructed by a Kronecker product of a LTE Rel-8 codebook and a unitary 2-by-2 matrix. Note that the idea described in 3GPP, R1-094752, “DL codebook design for 8Tx MIMO in LTE-A,” ZTE, RAN#59, Jeju, South Korea, November 2009 is to have a single codebook and the feedback is still a single index of the codebook.
As described in 3GPP, R1-094844, “Low-overhead feedback of spatial covariance matrix,” Motorola, RAN1#59, Jeju, South Korea, November 2009, a Kronecker product can be used for decomposing a bigger transmit covariance matrix R into two smaller matrices RULA and RPol, so that the feedback overhead can be reduced:R=RPolRULA  (3).
The above decomposition also works in the eigen-domain by applying the mixed-product property of a Kronecker product
                                                        R              =                            ⁢                                                R                  Pol                                ⊗                                  R                  ULA                                                                                                        =                            ⁢                                                [                                                            V                      Pol                                        ⁢                                          D                      Pol                                        ⁢                                          V                      Pol                      H                                                        ]                                ⊗                                  [                                                            V                      ULA                                        ⁢                                          D                      ULA                                        ⁢                                          V                      ULA                      H                                                        ]                                                                                                        =                            ⁢                                                                    [                                                                  V                        Pol                                            ⊗                                              V                        ULA                                                              ]                                    ⁡                                      [                                                                  D                        Pol                                            ⊗                                              D                        ULA                                                              ]                                                  ⁡                                  [                                                            V                      Pol                      H                                        ⊗                                          V                      ULA                      H                                                        ]                                                                                        (        4        )            where matrices “Vxx” contain the eigen-vectors of the transmit covariance matrices “Rxx”, respectively. Diagonal matrices “Dxx” contain the eigen-values of the transmit covariance matrices “Rxx”.
A key thing to point out is that the design principle of CSI feedback described in 3GPP, R1-094844, “Low-overhead feedback of spatial covariance matrixm,” Motorola, RAN1#59, Jeju, South Korea, November 2009 is to directly quantize the transmit covariance matrices, element-by-element wise. Such an approach is drastically different from the codebook-based quantization mentioned previously. So, even after Kronecker decomposition, the content of the feedback is still covariance matrix (or matrices), rather than codebook index (or indices).